where \(b\in \mathbb {R}^n\)  and \(A\in \mathbb {R}^{n\times n}\)  is a non-singular matrix. Starting from an initial guess \(x_0 \in \mathbb {R}^n\)  with residual \(\Phi (x_0) = b - Ax_0\) , GMRES [1]} generates a sequence \(\left\lbrace  x_i \right\rbrace \)  of approximate solutions to (REF ) with
\(x_i = \arg \min _{x} \left\lbrace  \Vert b - Ax\Vert _2 : x \in x_0 + \mathcal {K}_i \right\rbrace \,,\) 
